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Peer-to-Peer Networks 15 Self-Organization Christian Schindelhauer - - PowerPoint PPT Presentation

Peer-to-Peer Networks 15 Self-Organization Christian Schindelhauer Technical Faculty Computer-Networks and Telematics University of Freiburg Gnutella Connecting Protokoll - Ping Ping participants query for Ping neighbors Ping


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SLIDE 1

Peer-to-Peer Networks

15 Self-Organization

Christian Schindelhauer

Technical Faculty Computer-Networks and Telematics University of Freiburg

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SLIDE 2

Ping Ping Ping Ping Ping Ping Ping Pong Pong Pong Pong Pong Pong Pong

Gnutella — Connecting

  • Protokoll
  • Ping
  • participants query for

neighbors

  • are forwarded according

for TTL steps (time to live)

  • Pong
  • answers Ping
  • is forwarded backward on

the query path

  • reports IP and port adress

(socket pair)

  • number and size of

available files

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SLIDE 3

Degree Distribution in Gnutella

  • Modeling Large-scale Peer-to-Peer

Networks and a Case Study of Gnutella

  • Mihajlo A. Jovanovic, Master Thesis,

2001

  • The number of neighbors is

distributed according a power law (Pareto) distribution

  • log(#peers with degree d) = c - k log d
  • #peers with degree d = C/dk

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SLIDE 4

Pareto-Distribution Examples

  • Pareto 1897: Distribution of wealth in the

population

  • Yule 1944: frequency of words in texts
  • Zipf 1949: size of towns
  • length of molecule chains
  • file length of Unix-system files
  • ….

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SLIDE 5

Pareto Verteilung

  • Discreet Pareto-Distribution for x ∈ {1,2,3,…}
  • with constant factor
  • (also known as Riemann´s Zeta-function)
  • Heavy tail property
  • not all moments E[Xk] exist
  • the expectation exists if and only if (iff) α>2
  • variance and E[X2] exist iff α>3
  • E[Xk] exists iff α>k+1
  • Density function of the continuous function for x>x0

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SLIDE 6

Indegree and Outdegree of Web-Pages

  • are described by a power law (Pareto) distribution
  • Experiments of
  • Kumar et al 97: 40 millions Webpages
  • Barabasi et al 99: Domain *.nd.edu + Web-pages in distance 3
  • Broder et al 00: 204 millions web pages (Scan Mai und Okt. 1999)

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SLIDE 7

Connectivity of Pareto Graphs

  • William Aiello, Fan Chung, Linyuan Lu, A Random Graph

Model for Massive Graphs, STOC 2000

  • Undirected graph with n nodes where
  • the probability of k neighbors for a node is pk
  • where pk = c k-τ for some normalizing factor c
  • Theorem
  • For sufficient large n such Pareto-Graphs with exponent τ we observe
  • for τ < 1 the graph is connected with probability 1-o(1)
  • for τ > 1 the graph is nont connected with probability 1-o(1)
  • for 1< τ <2 there is a connected component of size Θ(n)
  • for 2< τ < 3.4785 there is only one connected component of size Θ(n) and

all others have size O(log n)

  • for τ >3.4785: there is no large connected component of size Θ(n) with

probability 1-o(1)

  • For τ >4: no large connected components which size can be described by a

power law (Pareto) distribution

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SLIDE 8

Zipf Distribution as a Variant of Power Laws

  • George Kinsley Zipf claimed
  • that the frequency of the n most frequent word f(n)
  • satisfies the equation n f(n) = c.
  • Zipf probability distribution for x ∈ {1,2,3,…}
  • with constant factor c only defined for connstant sized sets, since
  • is unbounded
  • Zipf distribution relate to the rank
  • The Zipf exponent α may be larger than 1, i.e. f(n) = c/nα
  • Pareto distribution realte the absolute size, e.g. the number of

inhabitants

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SLIDE 9

Size of towns Scaling Laws and Urban Distributions, Denise Pumain, 2003 Zipf distribution

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SLIDE 10

Zipf’s Law and the Internet

Lada A. Adamic, Bernardo A. Huberman, 2002 Pareto Distribution!!

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SLIDE 11

Heavy-Tailed Probability Distributions in the World Wide Web Mark Crovella, Murad, Taqqu, Azer Bestavros, 1996

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SLIDE 12

Small World Phenomenon

  • Milgram’s experiment 1967
  • 60 random chosen participants in Wichita, Kansas had to

send a packet to an unknown address

  • They were only allowed to send the packet to friends
  • likewise the friends
  • The majority of packtes arrived within six hops
  • Small-World-Networks
  • are networks with Pareto distributed node degree
  • with small diameter (i.e. O(logc n))
  • and relatively many cliques
  • Small-World-Networks
  • Internet, World-Wide-Web, nervous systems, social networks

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How do Small World Networks Come into Existence?

  • Emergence of scaling in random networks, Albert-Laszlo

Barabasi, Reka Albert, 1999

  • Preferential Attachment-Modell (Barabasi-Albert):
  • Starting from a small starting graph successively nodes are inserte with

m edges each (m is a parameter)

  • The probability to choose an existing node as a neighbor is proportional

to the current degree of a node

  • This leads to a Pareto network with exponent 2,9 ± 0,1
  • however cliques are very seldom
  • Watts-Strogatz (1998)
  • Start with a ring and connections to the m nearest neighbors
  • With probability p every edge is replaced with a random edge
  • Allows continuous transition from an ordered graph to chaos
  • Extended by Kleinberg (1999) for the theoretical verification of

Milgram‘s experiment

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SLIDE 14

Analyzing Gnutella

  • Modeling Large-scale Peer-to-Peer Networks and

a Case Study of Gnutella

  • Mihajlo A. Jovanovic, 2001

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SLIDE 15

Analyzing Gnutella

  • Modeling Large-scale Peer-to-Peer Networks and

a Case Study of Gnutella

  • Mihajlo A. Jovanovic, 2001
  • Comparison of the characteristic path length
  • mean distance between two nodes

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SLIDE 16

Peer-to-Peer Networks

15 Self-Organization

Christian Schindelhauer

Technical Faculty Computer-Networks and Telematics University of Freiburg